Algebraic matroids with graph symmetry

This paper studies the properties of two kinds of matroids: (a) algebraic matroids and
(b) finite and infinite matroids whose ground set have some canonical symmetry, for
example row and column symmetry and transposition symmetry.

For (a) algebraic matroids, we expose cryptomorphisms making them accessible to
techniques from commutative algebra. This allows us to introduce for each circuit in an
algebraic matroid an invariant called circuit polynomial, generalizing the minimal poly-
nomial in classical Galois theory, and studying the matroid structure with multivariate
methods.

For (b) matroids with symmetries we introduce combinatorial invariants capturing
structural properties of the rank function and its limit behavior, and obtain proofs
which are purely combinatorial and do not assume algebraicity of the matroid; these imply
and generalize known results in some specific cases where the matroid is also algebraic.
These results are motivated by, and readily applicable to framework rigidity, low-rank
matrix completion and determinantal varieties, which lie in the intersection of (a) and
(b) where additional results can be derived. We study the corresponding matroids and
their associated invariants, and for selected cases, we characterize the matroidal
structure and the circuit polynomials completely.